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# Direction Cosine Matrix Body to Wind to Alpha and Beta

Convert direction cosine matrix to angle of attack and sideslip angle

## Library

Utilities/Axes Transformations

## Description

The Direction Cosine Matrix Body to Wind to Alpha and Beta block converts a 3-by-3 direction cosine matrix (DCM) into angle of attack and sideslip angle. The DCM matrix performs the coordinate transformation of a vector in body axes (ox0, oy0, oz0) into a vector in wind axes (ox2, oy2, oz2). The order of the axis rotations required to bring this about is:

1. A rotation about oy0 through the angle of attack (α) to axes (ox1, oy1, oz1)

2. A rotation about oz1 through the sideslip angle (β) to axes (ox2, oy2, oz2)

Combining the two axis transformation matrices defines the following DCM.

To determine angles from the DCM, the following equations are used:

## Inputs and Outputs

InputDimension TypeDescription

First

3-by-3 direction cosine matrixTransforms body-fixed vectors to wind-fixed vectors.

OutputDimension TypeDescription

First

2-by-1 vectorContains angle of attack and sideslip angle, in radians.

## Assumptions and Limitations

This implementation generates angles that lie between ±90 degrees.

## Reference

Stevens, B. L., and F. L. Lewis, Aircraft Control and Simulation, John Wiley & Sons, New York, 1992.